**1. Introduction**

Northern elephant seals (*Mirounga angustirostris*) undergo regular periods of aphagia during their annual life cycle, as do many other phocids (Le Boeuf and Laws 1994). After nursing for about 30 days, the weaned pup fasts for 6-8 weeks, maintaining a fasting hyperglycemia, hyperlipidemia, hypoketonemia, and hypoinsulinemia (Champagne, *et al.*  2005). In most mammals, fasting is accompanied by hypoglycemia, and thus the fasting hyperglycemia in these animals is paradoxical. Previous studies of glucose metabolism in these animals (Keith and Ortiz 1989) indicate that the hyperglycemia results from both low rates of glucose utilization, due to very low insulin levels (Kirby, *et. al.* 1987), as well as high rates of glucose carbon recycling through both lactate and glycerol. Other studies indicate that fatty acids are the major energy substrate during this time (Castellini, *et. al.* 1987), and that these animals conserve nitrogen by having very low urea turnover and excretion rates (Houser and Costa 2001). Figure 1 shows a 10 compartment conceptual flow diagram of metabolite flux in fasting northern elephant seal pups as simulated in this study.

Mathematical models of biochemical systems are a prerequisite for a true understanding of the complexity of metabolic and physiologic systems. A model can be defined in both a physical and mathematical sense as a set of equations that describe the behavior of a dynamic system, and the response of the system to a given stimulus (Jeffers 1982). Many types of models exist. The models that most closely approximate reality are often the most complex, and it is often difficult to derive unbiased or valid estimates of model parameters. Matrix models offer a way to sacrifice some of the "reality" to gain the advantages of mathematical deduction and prediction (Jeffers 1978). Matrix models are ideally suited to simulate the results of isotope tracer experiments and linear compartment analysis models (Shipley and Clark 1972).

© 2012 Keith, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **2. Materials and methods**

The amount of carbon residing in each pool in Figure 1, and the fluxes between the pools, were determined using single injection radiotracer methods as described (Pernia, *et. al.* 1980, Keith and Ortiz 1989, and Castellini, *et. al.* 1987). If the animal is in steady state, the change in pool size (Q) for each compartment will be:

A Matrix Model of Fasting Metabolism in Northern Elephant Seal Pups 35

Where: Qj = Size of pool j (source of flux to pool i).


ji i.e. for j 1 10

If the coefficients of the differential equations (kij) are entered into a source/destination matrix (A), the matrix may be used to predict the magnitude of the flux between each

F = A \* Q0 Where F is the vector of the sum of the fluxes to and from all pools, Q0 is the vector of initial pool sizes, and A is the coefficient matrix derived above. The future state of the system then

Qt+1 = F \* Q0 Iteration of the above two operations allows computation, in difference equation format, of the future state of the system at any time desired. However, use of the matrix exponential permits analytical determination of the state of the system at any future time in one

Qt+1 = eAt \* Q0 Initially the system dynamics were simulated for an 8-week period, approximately as long as the duration of the fasting period of the northern elephant seal pups. In most cases, the simulation revealed monotonic (linear) declines in the sizes of the pools, with the obvious exception of the sink pool. However, in the case of the carbohydrate pools, i.e. glucose, glycerol and lactate, the pool sizes increased rapidly during the first part of the simulation, and then declined monotonically. This suggested that the estimates for the initial condition sizes of the pools were too low. In order to correct for this, a second, shorter (24 hr) simulation was conducted, after correcting the initial condition sizes of these pools by extrapolating the linear part of the pool size decline later in the simulation

A matrix may be described in terms of its characteristic equation, which will have the same order as the number of rows (= columns) in a square matrix (Jeffers 1978, Swartzman 1987). The roots of this characteristic equation are the eigenvalues (λ), which can be used to assess the stability of the system described by the matrix (Heinrich et al. 1977, Edelstein-Keshet 1988). Simply put, if all of the eigenvalues, or their real parts, are negative, the system is stable. If one or more eigenvalues are positive, the system is unstable. Zero value eigenvalues indicate a closed system (Edelstein-Keshet 1988, Halfon 1976, and Swartzman 1987). Other matrix parameters relevant to stability analysis are the trace (τ) and


compartment:

becomes:

operation:

back to time zero.

$$\mathbf{Q}^{(t)} = \mathbf{Q}^{(0)} \mathbf{e}^{\mathrm{-kt}}$$

where k is the fractional turnover rate, t is time, and e is the base of the natural logarithms. These parameters may be estimated by injecting a known amount of tracer (q) labeled in some way (such as 14C or 3H) but which is metabolically indistinguishable from the metabolite of interest (tracee).

Blood samples are taken over time, and the specific activity of the tracer determined and plotted on semi-log paper. If the assumption of first order kinetics holds, the plot will be linear, and the slope of the line is k. The assumption of instantaneous mixing allows extrapolation of the line back to time = 0, and the estimation of the specific activity at time = 0 (SA0). The size of the pool (Q) can then be estimated (Katz, *et. al.* 1974):

$$\mathbf{Q} = \frac{\mathbf{q}}{\mathbf{S} \mathbf{A}\_0}$$

The magnitude of the entry rate (R0 ) can then be estimated:

$$\mathbf{R}\_0 = \mathbf{k} \, \ast \, \mathbf{Q}$$

If the specific activity curve is plotted on regular paper and integrated, the Stewart-Hamilton equation provides a stochastic estimate of the irreversible loss rate (L) (Shipley and Clark 1972):

$$\mathbf{L} = \frac{\mathbf{q}}{\inf\_{\text{inf}}}$$

$$\int\_0 \mathbf{SA(t)dt}$$

Recycling rate (R) is the part of the entry rate (R0) which leaves the pool of interest and returns to it during the experiment. It is the difference between the entry rate (R0) and the irreversible loss rate (L) (Nolan and Leng 1974):

$$\mathbf{R} = \mathbf{R}\_0 \mathbf{\cdot} \mathbf{L}$$

Once the sizes of the pools and the flux rates between them are known, differential equations can be written to describe the rate of change of each compartment (Shipley and Clark 1972):

$$\text{dQ}\_{\text{i}}/\text{dt} = \left(\sum\_{\text{j}=1}^{10} \text{Q}^{\*} \text{ k}\_{\text{ij}}\right) + \left(\text{Q}\_{\text{i}}^{\*} \text{ k}\_{\text{ii}}\right) \text{for i} = 1 \rightarrow 10$$

Where: Qj = Size of pool j (source of flux to pool i).


34 New Approaches to the Study of Marine Mammals

in pool size (Q) for each compartment will be:

The amount of carbon residing in each pool in Figure 1, and the fluxes between the pools, were determined using single injection radiotracer methods as described (Pernia, *et. al.* 1980, Keith and Ortiz 1989, and Castellini, *et. al.* 1987). If the animal is in steady state, the change

Q(t) = Q(0)e-kt where k is the fractional turnover rate, t is time, and e is the base of the natural logarithms. These parameters may be estimated by injecting a known amount of tracer (q) labeled in some way (such as 14C or 3H) but which is metabolically indistinguishable from the

Blood samples are taken over time, and the specific activity of the tracer determined and plotted on semi-log paper. If the assumption of first order kinetics holds, the plot will be linear, and the slope of the line is k. The assumption of instantaneous mixing allows extrapolation of the line back to time = 0, and the estimation of the specific activity at time =

0

<sup>q</sup> Q SA

R0 = k \* Q If the specific activity curve is plotted on regular paper and integrated, the Stewart-Hamilton equation provides a stochastic estimate of the irreversible loss rate (L) (Shipley

inf

SA(t)dt

<sup>q</sup> <sup>L</sup>

 

0

Recycling rate (R) is the part of the entry rate (R0) which leaves the pool of interest and returns to it during the experiment. It is the difference between the entry rate (R0) and the

R = R0 - L Once the sizes of the pools and the flux rates between them are known, differential equations can be written to describe the rate of change of each compartment (Shipley and

<sup>10</sup>

dQ / dt ( Q \* k ) Q \* k for i 1 10

<sup>j</sup> <sup>i</sup> ij i ii j 1

0 (SA0). The size of the pool (Q) can then be estimated (Katz, *et. al.* 1974):

The magnitude of the entry rate (R0 ) can then be estimated:

irreversible loss rate (L) (Nolan and Leng 1974):

**2. Materials and methods** 

metabolite of interest (tracee).

and Clark 1972):

Clark 1972):


$$\text{i.e.}\_{\vec{\mu}} \text{for } \mathbf{j} = 1 \to 10$$

If the coefficients of the differential equations (kij) are entered into a source/destination matrix (A), the matrix may be used to predict the magnitude of the flux between each compartment:

$$\underline{\mathbf{F}} = \underline{\mathbf{A}}^\* \underline{\mathbf{Q}}$$

Where F is the vector of the sum of the fluxes to and from all pools, Q0 is the vector of initial pool sizes, and A is the coefficient matrix derived above. The future state of the system then becomes:

$$\mathbf{Q}^{\*1} = \mathbf{\bar{F}}^{\*} \mathbf{Q}^{0}$$

Iteration of the above two operations allows computation, in difference equation format, of the future state of the system at any time desired. However, use of the matrix exponential permits analytical determination of the state of the system at any future time in one operation:

$$\mathbf{Q}\_{\mathbb{H}^{+}} = \mathbf{e}\_{\mathbb{H}} \ast \mathbf{Q}^{\mathbb{H}}$$

Initially the system dynamics were simulated for an 8-week period, approximately as long as the duration of the fasting period of the northern elephant seal pups. In most cases, the simulation revealed monotonic (linear) declines in the sizes of the pools, with the obvious exception of the sink pool. However, in the case of the carbohydrate pools, i.e. glucose, glycerol and lactate, the pool sizes increased rapidly during the first part of the simulation, and then declined monotonically. This suggested that the estimates for the initial condition sizes of the pools were too low. In order to correct for this, a second, shorter (24 hr) simulation was conducted, after correcting the initial condition sizes of these pools by extrapolating the linear part of the pool size decline later in the simulation back to time zero.

A matrix may be described in terms of its characteristic equation, which will have the same order as the number of rows (= columns) in a square matrix (Jeffers 1978, Swartzman 1987). The roots of this characteristic equation are the eigenvalues (λ), which can be used to assess the stability of the system described by the matrix (Heinrich et al. 1977, Edelstein-Keshet 1988). Simply put, if all of the eigenvalues, or their real parts, are negative, the system is stable. If one or more eigenvalues are positive, the system is unstable. Zero value eigenvalues indicate a closed system (Edelstein-Keshet 1988, Halfon 1976, and Swartzman 1987). Other matrix parameters relevant to stability analysis are the trace (τ) and

determinant (Δ) of the matrix. The trace is the sum of the eigenvalues, and the determinant in the product of the eigenvalues (Heinrich et al. 1977). A dimensionless τ-Δ parameter plane may be envisioned which relates the magnitude of these two parameters to model stability, or the type of instability (Edelstein-Keshet 1988). All matrix calculations described herein were conducted using MATLAB.

A Matrix Model of Fasting Metabolism in Northern Elephant Seal Pups 37

 Qi FLUX 1. Adipose 1.2500e+002 -1.2750e-003 2. Glycerol 2.3100e-0 01 1.3800e-005 3. Palmitate 6.0000e-003 1.5000e-006 4. Lactate 7.5000e-003 3.9000e-005 5. Glucose 4.0000e-002 4.6722e-004 6. Alanine 4.4000e-003 -2.4304e-004 7. Urea 4.1700e-001 -1.2070e-004 8. Protein 2.0000e+002 -5.6160e-005 9. Ketones 3.5000e-003 8.1250e-006 10. Sink 1.0000e-000 9.3508e-004

**Table 1.** Initial condition values in moles of carbon for the 10 compartment model shown in Figure 1.

SOURCE POOL

POOL **ADIPOSE GLYCEROL PALMITATE LACTATE GLUCOSE ALANINE UREA PROTEIN KETONES SINK** 

ADIPOSE -1.02e-5 0 0 0 0 0 0 0 0 0 GLYCEROL 5.10e-6 -2.70e-3 0 0 0 0 0 0 0 0 PALMITATE 5.10e-6 0 -1.06e-1 0 0 0 0 0 0 0 LACTATE 0 0 0 -1.20e-3 1.20e-3 0 0 0 0 0 GLUCOSE 0 2.70e-3 0 1.20e-3 -4.50e-3 3.30e-3 0 0 0 0 ALANINE 0 0 0 0 0 -6.80e-2 0 2.808e-7 0 0 UREA 0 0 0 0 0 1.00e-3 -3.00e-4 0 0 0 PROTEIN 0 0 0 0 0 0 0 -2.808e-7, 0 0 KETONES 0 0 3.25e-3 0 0 0 0 0 -3.25e-3 0 SINK 0 0 1.03e-1 0 3.30e-3 3.40e-2 3.00e-4 0 3.25e-3 -1 **Table 2.** Source/destination matrix used to simulate the kinetics of metabolite flux in fasting northern elephant seal pups. Non-diagonal elements represent the rate constants for flow from the column pool to the row pool, in terms of time-1. The diagonal elements are the sum all rate constants for flows from

The roots of this equation are the eigenvalues (λ) of the matrix: 0.00; -8.10x10-4; -4.89x10-3 ; -2.70x10-3; -3.25x10-3; -1.06x10-1; -1.02x10-5; -3.00x10-4; -6.80x10-2; -2.81x10-7. Notice that there is one zero value, indicating that this is a closed system. The remainder of the eigenvalues are all negative, indicating a stable system. The trace (τ), or sum of the eigenvalues, of matrix A is -0.186 and the determinant (Δ), or product of the eigenvalues of matrix A is zero. Figure 2 shows the dimensionless τ-Δ parameter plane. The point for matrix A would lie on the yaxis, just below the x-y intercept, indicating that the system approaches a saddle point of stability. Notice that τ2 is 0.0346, which is greater than 4Δ, which is zero, indicating again that the matrix lies in the portion of the phase-plane corresponding to a saddle point

The flux is the sum of the flows into and out of each pool during the first time step.

DESTINATION

the pool, i.e. the sum of the column.

condition (Heinrich et al. 1977).

**Figure 1.** A 10-compartment flow diagram of metabolite flux in fasting northern elephant seals as simulated in this study. The boxes (or pools) represent the moles of carbon present in the animal as each metabolite, and the arrows represent the interchange of carbon between metabolites. There are no inputs to the model because the animal is fasting.
